December 2024 Stereographic Markov chain Monte Carlo
Jun Yang, Krzysztof Łatuszyński, Gareth O. Roberts
Author Affiliations +
Ann. Statist. 52(6): 2692-2713 (December 2024). DOI: 10.1214/24-AOS2426

Abstract

High-dimensional distributions, especially those with heavy tails, are notoriously difficult for off-the-shelf MCMC samplers: the combination of unbounded state spaces, diminishing gradient information, and local moves results in empirically observed “stickiness” and poor theoretical mixing properties—lack of geometric ergodicity. In this paper, we introduce a new class of MCMC samplers that map the original high-dimensional problem in Euclidean space onto a sphere and remedy these notorious mixing problems. In particular, we develop random-walk Metropolis type algorithms as well as versions of the Bouncy Particle Sampler that are uniformly ergodic for a large class of light and heavy-tailed distributions and also empirically exhibit rapid convergence in high dimensions. In the best scenario, the proposed samplers can enjoy the “blessings of dimensionality” that the convergence is faster in higher dimensions.

Funding Statement

JY was supported by Florence Nightingale Fellowship, Lockey Fund, and St Peter’s College Research Fund (O’Connor Fund) from University of Oxford.
KŁ was supported by a Royal Society University Research Fellowship.
GOR was supported by EPSRC grants Bayes for Health (R018561), CoSInES (R034710) PINCODE (EP/X028119/1), and EP/V009478/1. He also acknowledges financial support provided by the UKRI grant, EP/Y014650/1 as part of the ERC Synergy project OCEAN.

Citation

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Jun Yang. Krzysztof Łatuszyński. Gareth O. Roberts. "Stereographic Markov chain Monte Carlo." Ann. Statist. 52 (6) 2692 - 2713, December 2024. https://doi.org/10.1214/24-AOS2426

Information

Received: 1 March 2023; Revised: 1 February 2024; Published: December 2024
First available in Project Euclid: 18 December 2024

Digital Object Identifier: 10.1214/24-AOS2426

Subjects:
Primary: 60J20 , 60J25 , 65C05

Keywords: blessings of dimensionality , heavy tailed distributions , Piecewise deterministic Markov processes , Random walk Metropolis , stereographic projection , uniform ergodicity

Rights: Copyright © 2024 Institute of Mathematical Statistics

Vol.52 • No. 6 • December 2024
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